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Mirrors > Home > ILE Home > Th. List > ialgrlem1st | GIF version |
Description: Lemma for ialgr0 12076. Expressing algrflemg 6255 in a form suitable for theorems such as seq3-1 10491 or seqf 10492. (Contributed by Jim Kingdon, 22-Jul-2021.) |
Ref | Expression |
---|---|
ialgrlem1st.f | ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) |
Ref | Expression |
---|---|
ialgrlem1st | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algrflemg 6255 | . . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹‘𝑥)) | |
2 | 1 | adantl 277 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹‘𝑥)) |
3 | ialgrlem1st.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) | |
4 | 3 | adantr 276 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐹:𝑆⟶𝑆) |
5 | simprl 529 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆) | |
6 | 4, 5 | ffvelcdmd 5673 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐹‘𝑥) ∈ 𝑆) |
7 | 2, 6 | eqeltrd 2266 | 1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ∘ ccom 4648 ⟶wf 5231 ‘cfv 5235 (class class class)co 5896 1st c1st 6163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fo 5241 df-fv 5243 df-ov 5899 df-1st 6165 |
This theorem is referenced by: ialgr0 12076 algrp1 12078 |
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